NEW HERE? LEARN MORE
closeclose
Last visit was: 25 Apr 2024, 04:11 It is currently 25 Apr 2024, 04:11
Decision Tracker
My Rewards
New posts
New comers' posts

Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

What is the range of the roots of ||x 1| 2| = 1?

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 92912
Own Kudos [?]: 618906 [44]
Given Kudos: 81595
Send PM
CAT Tests Forum Quiz Mode
What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   29 Aug 2017, 02:02
6
Kudos
38
Bookmarks
Expert Reply
00:00
A
B
C
D
E

Difficulty:

45% (medium)

Question Stats:

66% (01:46) correct 34% (01:45) wrong based on 972 sessions
Hide Show timer Statistics

Fresh GMAT Club Tests' Challenge Question:



What is the range of the roots of \(||x – 1| – 2| = 1\)?

A. 0
B. 2
C. 4
D. 6
E. 8


M37-17

Experience a GMAT Club Test Questions
Yes, you've landed on a GMAT Club Tests question
Craving more? Unlock our full suite of GMAT Club Tests here
Want to experience more? Get a taste of our tests with our free trial today
Rise to the challenge with GMAT Club Tests. Happy practicing!
GMAT Club Tests Check Our Products Try Free Tests
ShowHide Answer
Official Answer
D
Most Helpful Reply
User avatar
L
Mo2men
RSM Erasmus Moderator
Joined: 26 Mar 2013
Posts: 2461
Own Kudos [?]: 1360 [15]
Given Kudos: 641
Concentration: Operations, Strategy
Schools: Erasmus (II)
Send PM
Reviews Badge
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   30 Aug 2017, 03:52
12
Kudos
3
Bookmarks
Bunuel wrote:
What is the range of the roots of \(||x – 1| – 2| = 1\)?

A. 0
B. 2
C. 4
D. 6
E. 8


Another approach

Let \(z = x -1\)

\(||z| – 2| = 1\).............square both sides

\(z^2-4|z|+4=1\)

\(z^2-4|z|+3=0\)

\((|z|-3)(|z|-1)=0\)

\(|z|=3\) or \(|z|=1\)

substitute z from above

\(x–1=3\) or \(x–1=-3\)
\(x=4\) or \(x=-2\)

OR

\(x–1=1\) or \(x–1=-1\)
\(x=2\) or \(x=0\)

Range = Max value - min value

\(R=4-(-2)=4+2=6\)

Answer: D
User avatar
P
Luckisnoexcuse
Current Student
Joined: 18 Aug 2016
Posts: 531
Own Kudos [?]: 577 [9]
Given Kudos: 198
Concentration: Strategy, Technology
Schools: Kenan-Flagler '20 (M)
GMAT 1: 630 Q47 V29
GMAT 2: 740 Q51 V38
Send PM
GMAT ToolKit User Reviews Badge
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   29 Aug 2017, 02:44
4
Kudos
5
Bookmarks
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the range of the roots of \(||x – 1| – 2| = 1\)?

A. 0
B. 2
C. 4
D. 6
E. 8


||x – 1| – 2| = 1
|x – 1| – 2 = 1 or |x – 1| – 2 = -1

if |x – 1| – 2 = 1
|x – 1| = 3
x – 1 = 3 or x – 1 = -3
giving us x = 4, -2

if |x – 1| – 2 = -1
|x – 1| = 1
x – 1 = 1 or x – 1 = -1
giving us x = 2, 0

now we have x=-2,0,2,4
range will be 4+2 = 6
D
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 92912
Own Kudos [?]: 618906 [2]
Given Kudos: 81595
Send PM
CAT Tests Forum Quiz Mode
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   19 Nov 2021, 04:46
2
Bookmarks
Expert Reply
Official Solution:

What is the range of the roots of \(||x – 1| – 2| = 1\)?

A. \(0\)
B. \(2\)
C. \(4\)
D. \(6\)
E. \(8\)


\(||x – 1| – 2| = 1\)

\(|x – 1| – 2 = 1\) or \(||x – 1| – 2| = -1\)

CASE 1:

\(|x – 1| – 2 = 1\)

\(|x – 1| = 3\)

\(x=4\) or \(x=-2\).

CASE 2:

\(|x – 1| – 2 = -1\)

\(|x – 1| = 1\)

\(x=2\) or \(x=0\).

So, \(x\) can be -2, 0, 2, and 4. The range of values \(= 4-(-2)=6\).


Answer: D
General Discussion
User avatar
L
pushpitkc
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 2873
Own Kudos [?]: 5205 [2]
Given Kudos: 47
Location: India
GPA: 3.12
Send PM
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   31 Aug 2017, 20:22
1
Kudos
The expression is in the form ||a| - 2| = 1

There are 4 possibilities, when trying to solve this equation
|-a -2| = 1
There are 2 options:
-(-a -2) = 1 => a +2 = 1 => a = -1
(-a - 2) = 1 => -a = 3 => a = -3

|a - 2| = 1
There are 2 options:
-(a - 2) = 1 => -a + 2 = 1 => a = 1
(a - 2) = 1 => a = 3

Hence, there are 4 values for a which are -3,-1,1,3

Coming back to the question, if we substitute a to be x-1

x - 1 = -3 => x = -2
x - 1 = -1 => x = 0
x - 1 = 1 => x = 2
x - 1 = 3 => x = 4

Hence, the range of the values is 6(Option D)
User avatar
G
nkmungila
Manager
Manager
Joined: 07 Jun 2017
Posts: 116
Own Kudos [?]: 672 [1]
Given Kudos: 60
Location: India
Concentration: Technology, General Management
GMAT 1: 660 Q46 V38
GPA: 3.6
WE:Information Technology (Computer Software)
Send PM
GMAT ToolKit User Reviews Badge
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   31 Aug 2017, 22:20
1
Bookmarks
the values are
X-3=1
-X+3 = 1
-X-1 = 1
X+1 = 1

the values X are { 4, 2, 0, -2}

hence range = 4-(-2)
6
Answer D
avatar
B
Darknight2
Intern
Intern
Joined: 06 Apr 2016
Posts: 21
Own Kudos [?]: 11 [3]
Given Kudos: 79
Location: India
Schools: Desautels '21
GMAT 1: 720 Q49 V40
Send PM
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   02 Sep 2017, 00:34
3
Bookmarks
||x-1|-2|=1

Removing the outer modulus, we get
|x-1|-2 = 1 (or) -1

|x-1|= 3 (or) 1

Now, if we remove the modulus for |x-1| we will get the following four possible values for x

x-1= 3 (or) -3 => x= 4 (or) -2
X-1= 1 (or) -1 => x= 2 (or) 0

Hence the four possible values of x are (-2,0,2,4)
=> Range= 4- (-2) = 6

Ans-> Option D
avatar
B
laulau
Intern
Intern
Joined: 09 Apr 2018
Posts: 26
Own Kudos [?]: 28 [2]
Given Kudos: 54
GPA: 4
Send PM
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   17 Sep 2018, 03:54
2
Kudos
I just applied a rule for complex absolute value equations:

If the equation contains 1 variable and 1 or more than 1 constant(s) in 1 or more than 1 absolute value expressions, the only two cases we have to consider are (1) that the expressions have the same sign and (2) that they have different signs.

Applied to the question here I computed the following:

case (1):

(x-1)-2=1
x-3=1
x=4

case (2)
-(x-1)-2=1
-x+1-2=1
-x-1=1
-2=x

Since we are asked for the range (= highest value-lowest value), we have to perform this final step to get to the answer: 4-(-2)=6.

Please hit Kudos if you liked this approach :)
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 92912
Own Kudos [?]: 618906 [0]
Given Kudos: 81595
Send PM
CAT Tests Forum Quiz Mode
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   24 Dec 2018, 04:53
Expert Reply
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the range of the roots of \(||x – 1| – 2| = 1\)?

A. 0
B. 2
C. 4
D. 6
E. 8


Par of GMAT CLUB'S New Year's Quantitative Challenge Set

User avatar
L
Kinshook
GMAT Club Legend
GMAT Club Legend
Joined: 03 Jun 2019
Posts: 5343
Own Kudos [?]: 3964 [2]
Given Kudos: 160
Location: India
Schools: HBS '22 CBS '22 Wharton '22
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Send PM
Reviews Badge
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   Updated on: 08 Oct 2019, 04:00
1
Kudos
1
Bookmarks
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the range of the roots of \(||x – 1| – 2| = 1\)?

A. 0
B. 2
C. 4
D. 6
E. 8


Case 1: |x-1|-2 = 1
|x-1| = 3
x = 4 or -2

Case 2: |x-1|-2 = -1
|x-1| = 1
x = 2 or 0

Set of roots x = {-2,0,2,4}
Range of roots = 4 - (-2) = 6

IMO D

Originally posted by Kinshook on 08 Oct 2019, 03:13.
Last edited by Kinshook on 08 Oct 2019, 04:00, edited 1 time in total.
User avatar
D
satya2029
Manager
Manager
Joined: 10 Dec 2017
Posts: 235
Own Kudos [?]: 207 [0]
Given Kudos: 135
Location: India
Send PM
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   08 Oct 2019, 03:31
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the range of the roots of \(||x – 1| – 2| = 1\)?

A. 0
B. 2
C. 4
D. 6
E. 8


lx-1l=y
ly-2l=1
y-2=1 or y-2=-1
y=3 or y=1
lx-1l=3 or lx-1l=1
x-1=3 or x-1=-3 or x-1=1 or x-1=-1
x=4 or x=-2 or x=2 or x=0
x=-2 smallest
x=4 largest
range=4-(-2)
D:)
User avatar
L
nick1816
Retired Moderator
Joined: 19 Oct 2018
Posts: 1878
Own Kudos [?]: 6296 [0]
Given Kudos: 704
Location: India
Send PM
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   08 Oct 2019, 03:56
Slope of each line is 1 or -1

Hence the range of the roots of \(||x – 1| – 2| = 1\)= 1+1+1+1+1+1=6 (clearly seen from the graph)

Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the range of the roots of \(||x – 1| – 2| = 1\)?

A. 0
B. 2
C. 4
D. 6
E. 8

Attachments

Untitled.png
Untitled.png [ 4.78 KiB | Viewed 13649 times ]

User avatar
D
exc4libur
SVP
SVP
Joined: 24 Nov 2016
Posts: 1720
Own Kudos [?]: 1344 [0]
Given Kudos: 607
Location: United States
Send PM
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   11 Nov 2019, 08:51
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the range of the roots of \(||x – 1| – 2| = 1\)?

A. 0
B. 2
C. 4
D. 6
E. 8


|x-1|≥0 when x≥1 (positive); |(x-1)-2|=1…|x-3|=1;
|x-3|≥0 when x≥3: (x-3)=1…x=4=valid (x≥3)
|x-3|<0 when x<3: -(x-3)=1…-x+3=1…x=2=valid (x<3)

|x-1|<0 when x≤1 (negative); |-(x-1)-2|=1…|-x-1|=1
|-x-1|≥0 when x≤-1 (positive): (-x-1)=1…-x=2…x=-2=valid (x≤-1)
|-x-1|<0 when x>-1 (negative): -(-x-1)=1…x+1=1…x=0=valid (x>-1)

valid solutions: x={4,2,-2,0} range=largest-smallest=4-(-2)=6

Ans (D)
User avatar
S
kaladin123
Current Student
Joined: 06 Jul 2019
Posts: 135
Own Kudos [?]: 59 [0]
Given Kudos: 747
Location: India
Concentration: General Management, Strategy
Schools: IIMA PGPX'23 (II)
GMAT 1: 640 Q39 V39
GMAT 2: 700 Q48 V38 (Online)
GPA: 3.11
WE:Project Management (Computer Software)
Send PM
Reviews Badge
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   28 Aug 2021, 11:59
Alternative Approach
If \(a\) & \(b\) are two numbers on the number line, \(|a-b|\) represents the absolute value of the difference between them.
Likewise, we can draw a couple of number lines to infer that \(x=\{0,2,-2,4\}\)
Attachment:
nl3.jpg
nl3.jpg [ 19.6 KiB | Viewed 9944 times ]


Hence, \(Range=Max-Min=4-(-2)=6\)
User avatar
P
Crytiocanalyst
Director
Director
Joined: 16 Jun 2021
Posts: 994
Own Kudos [?]: 183 [0]
Given Kudos: 309
Send PM
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink] New post   31 Aug 2021, 00:58
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the range of the roots of \(||x – 1| – 2| = 1\)?

A. 0
B. 2
C. 4
D. 6
E. 8

– 1| – 2| = 1
|x – 1| – 2 = 1 or |x – 1| – 2 = -1

|x – 1| – 2 = 1
|x – 1| = 3
x – 1 = 3 or x – 1 = -3
Therefore x = 4, -2

if |x – 1| – 2 = -1
|x – 1| = 1
x – 1 = 1 or x – 1 = -1
x = 2, 0


range equals 4+2 = 6

Therefore IMO D
GMAT Club Bot
gmatclubot
Re: What is the range of the roots of ||x 1| 2| = 1? [#permalink]
31 Aug 2021, 00:58
Moderators:
Bunuel
Math Expert
92912 posts
yashikaaggarwal
Senior Moderator - Masters Forum
3137 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions